TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.19.5.122
TI  - Geometric expansion of fluctuations and average shadows
PY  - 2025/11/07
UR  - https://www.scipost.org/SciPostPhys.19.5.122
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 19
IS  - 5
SP  - 122
A1  - Berthière, Clément
AU  - Estienne, Benoit
AU  - Stéphan, Jean-Marie
AU  - Witczak-Krempa, William
AB  - Fluctuations of observables provide unique insights into the nature of physical systems, and their study stands as a cornerstone of both theoretical and experimental science. Generalized fluctuations, or cumulants, provide information beyond the mean and variance of an observable. In this paper, we develop a systematic method to determine the asymptotic behavior of cumulants of local observables as the region becomes large. Our analysis reveals that the expansion is closely tied to the geometric characteristics of the region and its boundary, with coefficients given by convex moments of the connected correlation function: the latter is integrated against intrinsic volumes of convex polytopes built from the coordinates, which can be interpreted as average shadows. A particular application of our method shows that, in two dimensions, the leading behavior of odd cumulants of conserved quantities is topological, specifically depending on the Euler characteristic of the region. We illustrate these results with the paradigmatic strongly-interacting system of two-dimensional quantum Hall state at filling fraction $1/2$, by performing Monte-Carlo calculations of the skewness (third cumulant) of particle number in the Laughlin state.
ER  -

@Article{10.21468/SciPostPhys.19.5.122,
	title={{Geometric expansion of fluctuations and average shadows}},
	author={Clément Berthière and Benoit Estienne and Jean-Marie Stéphan and William Witczak-Krempa},
	journal={SciPost Phys.},
	volume={19},
	pages={122},
	year={2025},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.19.5.122},
	url={https://scipost.org/10.21468/SciPostPhys.19.5.122},
}